Event Studies

Simple Panel Data Approaches with Binary Treatment

Vladislav Morozov

INTRODUCTION

Learning Outcomes

Textbook References

Further Reading

Reference articles:

  • MacKinlay (1997)
  • Freyaldenhoven et al. (2021)
  • Miller (2023)

Latter two articles more advanced and modern — how people do it these days.

Also see NBER 2023 Methods Lectures on Linear Panel Event Studies: https://www.nber.org/conferences/si-2023-methods-lectures-linear-panel-event-studies

TWO PERIODS

SETTING

Simple Event Study Setting

Begin with the simplest possible panel setting with binary treatment:

  • Two periods with \(N\) units:
    • No treatment in period 1.
    • All units treated in period 2.
  • Data: outcomes \((Y_{i1}, Y_{i2})\).

Object of interest: “average effect of treatment”

Simple Estimator — Average Change

Simplest approach: compute average change in \(Y_{it}\) across periods \[ \widehat{AE}_{ES} = \dfrac{1}{N}\sum_{i=1}^N (Y_{i2}- Y_{i1}). \tag{1}\]

Estimator (1) — simplest example of event study estimators (see Freyaldenhoven et al. 2021; Miller 2023).

Example Framework

Possible empirical framework

  • Units \(i\): firms that make phones
  • Outcome \(Y_{it}\): their stock price
  • Periods:
    1. One week before Apple announces the iPhone.
    2. One week after the announcement

Effect of interest: change in stock prices due to the announcement of iPhone

What Does (1) Do?

Proposition 1 (Asymptotics for \(\widehat{AE}_{ES}\)) Let

  • (Cross-sectional random sampling): \((Y_{i1}, Y_{i2})\) be independent and identically distributed (IID)
  • Finite first moments: \(\E[\abs{Y_{it}}]<\infty\)

Then \[ \widehat{AE}_{ES} \xrightarrow{p} \E[Y_{i2} - Y_{i1}]. \]

CAUSAL ANALYSIS

Causal Framework

Is \(\E[Y_{i2} - Y_{i1}]\) interesting (=causal)?

Need a causal framework to talk about causal effects!

Work in the familiar potential outcomes framework:

  • \(Y_{it}^0\) — outcome for \(i\) in period \(t\) if not treated
  • \(Y_{i1}^1\) — outcome for \(i\) in period \(t\) if treated
  • Treatment effect for \(i\) in \(t\): \(Y_{it}^1- Y_{it}^0\)

For short, use \(Y_{it}^d\) where \(d=0, 1\)

Limit of ES Estimator and Causality

Potential and realized outcomes are connected as \[ Y_{i2} = Y_{i2}^1, \quad Y_{i1} = Y_{i1}^0. \]

It follows that \[ \widehat{AE}_{ES} \xrightarrow{p} \E[Y_{i2}^2- Y_{i1}^1]. \]

\(\E[Y_{i2}^2- Y_{i1}^1]\) is not necessarily a treatment effect — mixes effect of treatment and effects of time!

Example

Context Again consider the iPhone example. Then

  • \(Y_{i2}^1 - Y_{i2}^0\) — treatment effect, change in price because of the iPhone announcement
  • \(Y_{i2}^0 - Y_{i0}^0\) — change in a world without iPhone

We see combination of both changes \[ Y_{i2} - Y_{i1} = Y_{i2}^1- Y_{i1}^0 = [Y_{i2}^1- Y_{i2}^0] + [Y_{i2}^0 - Y_{i0}^0] \]

Solution: Restrict Changes over Time

Simple solution: rule out changes over time

Assumption: no variation in potential outcomes \[ Y_{i2}^d= Y_{i1}^d, \quad d=0, 1 \]

Then \(\widehat{AE}_{ES}\) is estimating a causal parameter — average effects \[ \begin{aligned} \widehat{AE}_{ES} & \xrightarrow{p} \E[Y_{i1}^1- Y_{i1}^0] = \E[Y_{i2}^1- Y_{i2}^0] \end{aligned} \]

Summary so Far

REGRESSION INTERPRETATION

Regression Setting

Can also connect \(\widehat{AE}_{ES}\) and OLS

Consider regression model \[ \begin{aligned} Y_{it} & = \beta_0 + \beta_1 D_{it} + u_{it}, \\ D_{it} & = \begin{cases} 1, & t= 1 \\ 0, & t =0 \end{cases} \end{aligned} \tag{2}\] where we simply treat \((Y_{i1}, D_{i1})\) and \((Y_{i2}, D_{i2})\) as separate observations

Event Study and OLS

  1. Can use all results developed for OLS for \(\widehat{AE}_{ES}\)
  2. Regressing \(Y_{it}\) on \(D_{it}\) gives a causal parameter
    • Under no trends
    • No experiment was necessary

Event Study and Regression

A way to think about regression in causal settings:

  • Write down the regression in terms of parameters of interest: e.g. let \[ \beta_0 = \E[Y_{i0}^0], \quad \beta_1 = \E[Y_{i2}^1- Y_{i2}^0] \]

  • Connect regression to potential outcomes: what is \(u_{it}\) in terms of potential outcomes?

  • Check properties of this \(u_{it}\). If \(u_{it}\) is “nice”, apply OLS (or another method)

MULTIPLE PERIODS

ESTIMATION AND CAUSAL FRAMEWORK

Multiple Period Framework

  • Often have more than 2 periods of data
  • Want to use that data

New framework:

  • \(T\) periods in total
  • Treatment starts in period \(t_0\)
  • We see \(Y_{it}^0\) for \(t<t_0\) and \(Y_{it}^1\) for \(t\geq t_0\)

Expanded Regression

New variables for treatment: \[ D_{it, \tau} = \begin{cases} 1, & t= \tau, \\ 0, & t\neq \tau \end{cases} \]

Can try similar regression: \[ Y_{it} = \beta_0 + \sum_{\tau = t_0}^{T} \beta_\tau D_{it, \tau} + u_{it}. \tag{3}\]

Estimates

Fairly easy to show that \[ \begin{aligned} \hat{\beta}^{OLS}_{\tau} & = \dfrac{1}{N} \sum_{i=1}^N Y_{i\tau} - \dfrac{1}{N(t_0-1)} \sum_{i=1}^N\left[ Y_{i1} + \dots + Y_{it_0-1} \right] \\ & \xrightarrow{p} \E\left[Y_{i\tau}^1 - \dfrac{1}{t_0-1}(Y_{i1}^0+ \dots + Y_{it_0-1}^0) \right] \end{aligned} \] More general version of the simple estimator of before

Dynamic Treatment Effects?

If \(\beta_{\tau}\) — average effect in period \(\tau\), then model (3) seems to allow for dynamic effects


Dynamic effects often realistic: effect of treatment may grow or disappear over time. Example: impact of job training on earnings:

  • Disappearing: you forget the training over time
  • Increasing: job training lets you jump to a higher position and gain experience quicker for the rest of your life

Estimation of Dynamic Average Treatment Effects

Under the no trends in the baseline assumption: \[ \begin{aligned} & \E\left[Y_{i\tau}^1 - \dfrac{1}{t_01}(Y_{i1}^0+ \dots + Y_{it_0-1}^0) \right] = \E[Y_{i \tau}^1 - Y_{i\tau}^0] \end{aligned} \] Right hand side is average effect in period \(\tau\)

ASYMPTOTIC PROPERTIES

Consistency in the Multivariate Case

Unbiasedness of OLS

Moreover, under no trends in the baseline \[ \E[\hat{\beta}_{\tau}] = \E[Y_{i \tau}^1 - Y_{i\tau}^0] \] In other words, the OLS estimator is unbiased

Asymptotic Distribution

Asymptotic Variance I

Need variance \(V\) to construct confidence intervals for \(\E[Y_{i \tau}^1 - Y_{i\tau}^0]\). How to find \(V\)? Remember the CLT:

If \(X_1, X_2, \dots\) are IID random variables with \(\E[X_i]=\mu\) and \(\E[X^2]<\infty\), then \[ \sqrt{N}\left( \dfrac{1}{N}\sum_{i=1}^N X_i - \mu \right)\Rightarrow N\left(0, \var(X_i) \right) \] Asymptotic variance = variance of \(X_i\)

Asymptotic Variance II

Can apply this idea to find \(V\). Can write \[ \begin{aligned} \hat{\beta}^{OLS}_{\tau} & = \dfrac{1}{N} \sum_{i=1}^N Z_i\\ Z_i & = Y_{i\tau} - \dfrac{1}{(t_0-1)}\left[ Y_{i1} + \dots + Y_{it_0-1} \right] \end{aligned} \]

Then we get that \[ V = \var(Z_i) \]

Estimator for Asymptotic Variance

Can estimate \(V\) with \[ \hat{V} = \widehat{\var}(Z_i) = \dfrac{1}{N}\left(Z_i - \dfrac{1}{N}\sum_{j=1}^N Z_j \right)^2 \]

Estimated standard error of \(\hat{\beta}_{\tau}\): \[ \widehat{se}(\hat{\beta}_{\tau}) = \sqrt{ \dfrac{\hat{V}}{N} } \]

Inference on Average Effects I

Can now construct confidence intervals and hypothesis tests about \(\E[Y_{i \tau}^1 - Y_{i\tau}^0]\). E.g. an asymptotic 95% confidece interval: \[ \widehat{CI}_{95\%} = \left[\hat{\beta}^{OLS}_{\tau} - z_{1-\alpha/2}\widehat{se}(\hat{\beta}_{\tau}), \hat{\beta}^{OLS}_{\tau} + z_{1-\alpha/2}\widehat{se}(\hat{\beta}_{\tau}) \right] \] where the critical values \(z_{1-\alpha/2}\) come from the standard normal distribution \[ z_{1-\alpha/2}= \Phi^{-1}\left(1 - \dfrac{\alpha}{2} \right) \]

Inference on Average Effects II

But what if we want to set the joint hypothesis \[ H_0: \beta_{\tau} = 0, \quad \tau = t_0, \dots, T \]

  • This \(H_0\) is a hypothesis above a vector of effects
  • Need a joint asymptotic distribution to write down a test

Joint Asymptotic Distribution

Proposition 6 (Joint Asymptotics for Estimated Effects) Let

  • No trends in the baseline assumption hold
  • \((Y_{i1}, Y_{i2}, \dots, Y_{iT})\) be IID (over \(i\))
  • Finite second moments: \(\E[Y_{it}^2]<\infty\)

Then there exists some variance-covariance matrix \(\bV\) such that \[ \sqrt{N}(\hat{\bbeta} -\bbeta) \Rightarrow N(0, \bV) \]

Joint Inference on Average Effects

Can use Proposition 6 to create a Wald test

EMPIRICAL ILLUSTRATION

Slide

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References

Freyaldenhoven, Simon, Christian Hansen, Jorge Pérez Pérez, and Jesse Shapiro. 2021. “Visualization, Identification, and Estimation in the Linear Panel Event-Study Design.” w29170. Cambridge, MA: National Bureau of Economic Research. https://doi.org/10.3386/w29170.
Huntington-Klein, Nick. 2025. The Effect: An Introduction to Research Design and Causality. S.l.: Chapman and Hall/CRC.
MacKinlay, A. Craig. 1997. “Event Studies in Economics and Finance.” Journal of Economic Literature 35 (1): 13–39.
Miller, Douglas L. 2023. “An Introductory Guide to Event Study Models.” Journal of Economic Perspectives 37 (2): 203–30. https://doi.org/10.1257/jep.37.2.203.